Original: Tangentially, for a few years I've been looking for a Youtube video, I think by Mathologer [1], that explained (geometrically?) how the Golden Ratio was the limit of the continued fraction 1+1/(1+1/(1+1/(...))).
Anyone know what I'm talking about?
I know Mathologer had a conflict with his editor at one point that may have sown chaos on his channel.
I’m curious if the geometric approach has any edge-case benefits—like better numerical stability—or if it’s purely for elegance.
I sat down and worked it out. What do you know golden ratio.
Oh and this other number, -0.618. Anyone know what it's good for?
The closest thing I do related to the golden ratio is using the harmonic armature as a grid for my paintings.
.sidebar { flex: 1; }
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(https://wonger.dev/enjoyables on desktop / wide viewport)
And yes, for the people who get hung up on what the Old Masters did, it’s mostly armature grids and not the golden ratio!
But if your units follow a golden ratio progression, you just need to "concatenate" 2 consecutive units (2 measuring sticks) in order to find the third. And so on.
https://news.ycombinator.com/item?id=44077741
I don't have the energy to delve into this shit again, I found another antique site + ancient measurement system combo where the same link between 1/5, 1, π and phi are intertwined: https://brill.com/view/journals/acar/83/1/article-p278_208.x... albeit in a different fashion. + it was used to square the circle on top of the same remarkable approximation of phi as
5/6π - 1
phi^2 = phi + 1
0.2 * pseudo-phi^2 = 0.2 * (pseudo-phi + 1) = π/6
I'm depressed. I tried to sleep as long a possible, because when I woke up, within 3 seconds, I was back in hell. I want it to end, seriously, I can't stand it anymore.
Are we really upvoting this on HN? Truly the end times have come.
> In this work, I propose a rigorous approach of this kind on the basis of algorithmic information theory. It is based on a single postulate: that universal induction determines the chances of what any observer sees next. That is, instead of a world or physical laws, it is the local state of the observer alone that determines those probabilities. Surprisingly, despite its solipsistic foundation, I show that the resulting theory recovers many features of our established physical worldview: it predicts that it appears to observers as if there was an external world that evolves according to simple, computable, probabilistic laws. In contrast to the standard view, objective reality is not assumed on this approach but rather provably emerges as an asymptotic statistical phenomenon. The resulting theory dissolves puzzles like cosmology’s Boltzmann brain problem, makes concrete predictions for thought experiments like the computer simulation of agents, and suggests novel phenomena such as “probabilistic zombies” governed by observer-dependent probabilistic chances. It also suggests that some basic phenomena of quantum theory (Bell inequality violation and no-signalling) might be understood as consequences of this framework.
You're welcome
We did some statistical analysis on the golden ratio and its use in art. It does indeed seem that artists gravitate away from regular geometry such as squares, thirds etc and towards recursive geometry such as the golden ratio and the root 2 rectangle. Most of our research was on old master paintings, so it might be argued that this was learned behavior, however one of our experiments seems to show that this preference is also present in those without any knowledge of such prescribed geometries.
Golden ratio is very specific, whereas any proportional that is vaguely close to 1.5 (equivalently, 2:1) gets called out as an example of golden ratio.
The same tendency exists among wannabe-mathematician art critics who see a spiral and label it a logarithmic spiral or a Fibonacci spiral.
There's nothing particularly interesting about that; phi is (1 + √5)/2. All numbers composed of integers, addition, subtraction, multiplication, division, and square roots can be constructed by compass and straightedge.